How can I copy part of a texture to another part of the same texture?
Activate the Magic UV addon

After unwrapping the mesh

Select a single face and set the face with the uv texture.

Press U and you get the extra option with UV Magic Addon below

Select Copy/Paste UV -> Copy UV- Default

Then select the face you want to paste or face with similar texture on other 
side

Press U 

Select Copy/Paste UV -> Paste UV- Default